Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 3.5.3.2. Let $f: X \rightarrow Y$ be a continuous function between topological spaces. Then $f$ is a weak homotopy equivalence of topological spaces if and only if $\operatorname{Sing}_{\bullet }(f)$ is a weak homotopy equivalence of simplicial sets. This is a special case of Proposition 3.1.5.12, since the simplicial sets $\operatorname{Sing}_{\bullet }(X)$ and $\operatorname{Sing}_{\bullet }(Y)$ are Kan complexes (Proposition 1.1.9.8).